Application to the multiple prediction

Multiple prediction is the first step in the class of adaptive multiple suppression methods Verschuur et al. (1992). In a laterally homogeneous earth, Kelamis and Verschuur (2000) show that surface-related multiples can be predicted by taking the multi-dimensional auto-convolution of a common midpoint (CMP) gather. This auto-convolution reduces to a multiplication in the f-k domain, and so it can be performed rapidly with multi-dimensional FFT's.

Since multi-dimensional FFT's can be computed with a one-dimensional Fourier transform in helical coordinates, we can predict multiples by wrapping a CMP gather onto a helix, taking its 1-D FFT, squaring the result, and returning to the original domain.

We tested this algorithm on a single CMP from the synthetic BP multiple dataset Clapp (1999). Figure 6 displays the multiple prediction result using the helical coordinate system and only a single one-dimensional FFT. Theoretically, only first-order multiples should have correct relative amplitudes, and the source wavelet appears twice in the multiple prediction. However, the kinematics of all multiples are almost exact, even for higher-order multiples below 5 s two-way traveltime.

 

BP2
Figure 6 The left panel shows the multiple model obtained with the helix and a 1-D FFT. The right panel shows the input CMP gather with the offset axis reversed to facilitate the comparison. Some wrap-around effects appear at the top of the multiple model.